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Properties

Label 7.1.3.2a1.3-1.7.25a
Base 7.1.3.2a1.3
Degree \(7\)
e \(7\)
f \(1\)
c \(25\)

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Defining polynomial

$x^{7} + b_{20} \pi^{3} x^{6} + a_{19} \pi^{3} x^{5} + b_{22} \pi^{4} x + \pi$

Invariants

Residue field characteristic: $7$
Degree: $7$
Base field: 7.1.3.2a1.3
Ramification index $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $25$
Absolute Artin slopes: $[\frac{37}{18}]$
Swan slopes: $[\frac{19}{6}]$
Means: $\langle\frac{19}{7}\rangle$
Rams: $(\frac{19}{6})$
Field count: $98$ (complete)
Ambiguity: $1$
Mass: $294$
Absolute Mass: $98$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 7 }$ within this relative family, not the relative extension.

Indices of inseparability: $[19,0]$
Associated inertia: $[1,1]$
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 98

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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
7.1.21.39a1.197 $x^{21} + 7 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.198 $x^{21} + 7 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.199 $x^{21} + 7 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.200 $x^{21} + 7 x^{20} + 7 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.201 $x^{21} + 7 x^{20} + 7 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.202 $x^{21} + 7 x^{20} + 7 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.203 $x^{21} + 14 x^{20} + 7 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.204 $x^{21} + 14 x^{20} + 7 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.205 $x^{21} + 14 x^{20} + 7 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.206 $x^{21} + 21 x^{20} + 7 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.207 $x^{21} + 21 x^{20} + 7 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.208 $x^{21} + 21 x^{20} + 7 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.209 $x^{21} + 28 x^{20} + 7 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.210 $x^{21} + 28 x^{20} + 7 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.211 $x^{21} + 28 x^{20} + 7 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.212 $x^{21} + 35 x^{20} + 7 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.213 $x^{21} + 35 x^{20} + 7 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.214 $x^{21} + 35 x^{20} + 7 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.215 $x^{21} + 42 x^{20} + 7 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.216 $x^{21} + 42 x^{20} + 7 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.217 $x^{21} + 42 x^{20} + 7 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 3$ undefined
7.1.21.39a1.218 $x^{21} + 14 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.219 $x^{21} + 14 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.220 $x^{21} + 7 x^{20} + 14 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.221 $x^{21} + 7 x^{20} + 14 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.222 $x^{21} + 14 x^{20} + 14 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.223 $x^{21} + 14 x^{20} + 14 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.224 $x^{21} + 21 x^{20} + 14 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.225 $x^{21} + 21 x^{20} + 14 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.226 $x^{21} + 28 x^{20} + 14 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.227 $x^{21} + 28 x^{20} + 14 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.228 $x^{21} + 35 x^{20} + 14 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.229 $x^{21} + 35 x^{20} + 14 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.230 $x^{21} + 42 x^{20} + 14 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.231 $x^{21} + 42 x^{20} + 14 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 6$ undefined
7.1.21.39a1.232 $x^{21} + 21 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.233 $x^{21} + 21 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.234 $x^{21} + 21 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.235 $x^{21} + 7 x^{20} + 21 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.236 $x^{21} + 7 x^{20} + 21 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.237 $x^{21} + 7 x^{20} + 21 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.238 $x^{21} + 14 x^{20} + 21 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.239 $x^{21} + 14 x^{20} + 21 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.240 $x^{21} + 14 x^{20} + 21 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.241 $x^{21} + 21 x^{20} + 21 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.242 $x^{21} + 21 x^{20} + 21 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.243 $x^{21} + 21 x^{20} + 21 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.244 $x^{21} + 28 x^{20} + 21 x^{19} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.245 $x^{21} + 28 x^{20} + 21 x^{19} + 49 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
7.1.21.39a1.246 $x^{21} + 28 x^{20} + 21 x^{19} + 147 x + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{14} + 3 z^7 + 3,3 z + 2$ undefined
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